Multiphysics Simulation using Direct Coupled-Field … · Multiphysics Simulation using Direct Coupled-Field Element Technology Stephen Scampoli ANSYS, Inc. NAFEMS 2020 ... • A

Multiphysics Simulation using Direct Coupled-Field Element Technology

Stephen ScampoliANSYS, Inc.

NAFEMS 2020 Vision of Engineering Analysis and Simulation

Agenda

• Multiphysics Overview• Direct Coupled-Field Elements• Applications

– Piezoelectricity– Thermal-Electric Coupling– Electroelasticity– Structural-Thermal-Electric Coupling

• Conclusions• Questions and Answers

Thermoelectric Generator


Multiphysics - Why Consider It?

• You can’t afford to ignore coupled physics when:– The real world is a multiphysics world– Your design depends on coupled physical

phenomena– You are faced with small error margins– Physical testing is too costly

Innovative companies are moving beyond single physics analysis MEMS

Thermoelectric Actuator


Methods of Coupling Physics

• Direct Coupling– A single analysis

employing a coupled-field element containing all the necessary DOFs to solve the coupled-field problem.

• Load Transfer– Two or more

analysis are coupled by applying results from one analysis as loads in another analysis.

ThermalFluidsEmag

Structural

Direct Coupling• Element-level coupling• Highly coupled physics• Single model & mesh

Thermal

Fluids

Structural

Emag

Load Transfer• Sequential solution• Separate model & mesh• Separation of expertise


Direct Coupled-Field Element

• A finite element that couples the effects of interrelated physics within the element matrices or load vectors, which contain all necessary terms required for the coupled physics solution.

• Strong (matrix) or weak (sequential) coupling• Many industry applications

DOF: UX, UY, UZ, TEMP, VOLT, (AZ)


Strong (Matrix) Coupling

• Finite element matrix equation:

• The coupling is accounted for by the off-diagonal matrices [K12] and [K21].

• Provides for a coupled solution in one iteration for linear problems

• Examples: Piezoelectricity, Seebeck effect, thermoelasticity

[ ] [ ][ ] [ ]

{ }{ }

{ }{ }⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

2

1

2

1

2221

1211

FF

XX

KKKK


Weak (Sequential) Coupling

• Finite element matrix equation:

• The coupled effect is accounted for in the dependency of [K11] and {F1} on {X2} as well as [K22] and {F2} on {X1}.

• At least two iterations are required to achieve a coupled response.

• Examples: Joule heating, electric forces, piezoresistivity

{ }{ }( )[ ] [ ][ ] { }{ }( )[ ]

{ }{ }

{ }{ }( ){ }{ }{ }( ){ }⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

212

211

2

1

2122

2111

X,XFX,XF

XX

X,XK00X,XK


Applications for Coupled-Field Elements

• Piezoelectric – Transducers, resonators, sensors and actuators, vibration control

• Piezoresistive– Pressure sensors, strain gauges

• Thermal-electric– Wires, busbars, Peltier coolers

• Thermoelastic damping– MEMS resonators

• Electroelastic– Actuators, artificial muscle

• Coriolis Effect– Quartz angular velocity sensors


Piezoelectric Analysis

• In a piezoelectric analysis, the structural and electrostatic fields are coupled by the piezoelectric constants [e]:

{T} - stress vector{S} - elastic strain vector[c] - elastic stiffness

{D} - electric flux density{E} - electric field intensity[ ] - dielectric permittivityε

[e] - piezoelectric matrix

Structural Electric


Piezoelectric Finite Element Equations

• Strong (Matrix) Coupling

• U and V are strongly coupled piezoelectric matrices [KUV] and [KUV]T

• Symmetric Matrices• Static, Harmonic, Transient

UU UV UU UUT

UV VV VV

K K C 0U M 0 FU U K -K 0 -CV 0 0 QV V

⎧ ⎫ ⎧ ⎫⎡ ⎤ ⎡ ⎤⎧ ⎫ ⎧ ⎫⎡ ⎤+ + =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎩ ⎭ ⎩ ⎭⎣ ⎦⎣ ⎦⎣ ⎦ ⎩ ⎭ ⎩ ⎭

& &&

& &&

UU

VV

UV

UU

VV

UU

Element matrices:K - structural stiffnessK - dielectric permittivity K - piezoelectric couplingC - structural dampingC - dielectric dissipationM - mass


Piezoelectric Fan Example

• Piezoelectric Fan– Spot cooling– Low magnetic permeability– Driven by piezoelectric bimorph

Blade

Bimorph

FR4 Board

Finite Element Mesh

Two piezoelectric layers with opposing polarity.

Image courtesy of Piezo Systems, Inc.


Piezoelectric Fan Example

• Results– Fan driven at resonance– 120 Volts at 60 Hz

+VinFixed


• In a thermoelectric analysis, the thermal and electric fields are coupled by the Joule heat QJ and Seebeck coefficients [α]:

Thermal-Electric Analysis

{q} - heat fluxT - temperature (absolute)[K] - thermal conductivity

{J} - electric current density{E} - electric field intensity[ ] - electrical conductivityσ[ ] - Seebeck coefficientsα

J TQ {J} {E} dVV

= ∫

Thermal Electric


Thermal-Electric Finite Element Equation

• Matrix and/or Load Vector Coupling

• T an V Coupling– Matrix

• Seebeck KVT

– Load Vector• Joule QJ Load Vector • Peltier QP Load Vector

• Symmetric - Joule Heating• Unsymmetric - Seebeck

P JTT TT

VT VV VV

K 0 C 0T Q+Q +QT+ =K K 0 CV IV

⎧ ⎫ ⎧ ⎫⎡ ⎤ ⎡ ⎤⎧ ⎫⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎩ ⎭⎣ ⎦ ⎣ ⎦ ⎩ ⎭⎩ ⎭

&

&

TT

VV

VT

TT

VV

Element matrices:K - thermal conductivityK - electric conductivity K - Seebeck coupling C - specific heat dampingC - dielectric damping


Thermoelectric Cooler Example

• Thermoelectric Cooler– Thermal-electric coupling

• n-type and p-type semiconductors

20 Amps

Ground

Cold Side (Heat Absorbed)

Hot Side (Heat Rejected)

Image courtesy of Marlow Industries


Thermoelectric Cooler Example

• Thermoelectric Results– Thermal gradient in Peltier blocks

Hot Side (Heat Rejected)

Cold Side (Heat Absorbed)

Temperature

Electric Potential

Current Density


Electroelastic Analysis

• In an electroelastic analysis, structural and electrostatic fields are coupled by an electric force {Fe}:

e{F } [ ]dVM

V

σ= − ∇•∫

( )T T T1[ ] - Maxwell stress tensor = {E}{D} + {D}{E} - {D} {E}{ }2

{D} - electric flux density {E}

Mσ Ι

- electric field intensity { } - identity matrixΙ

Structural Electrostatic


Electroelastic Finite Element Formulation

• Load Vector (Weak) Coupling

• U and V are couple through the electric force load vector {Fe}

• Iterative Solution• Symmetric Matrix

eUU UU UU

VV VV

K 0 C 0U M 0 F+FU U 0 K 0 CV 0 0 QV V

⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎡ ⎤ ⎡ ⎤⎧ ⎫ ⎡ ⎤+ + =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎩ ⎭ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎩ ⎭⎩ ⎭ ⎩ ⎭

& &&

& &&

UU

VV

UU

VV

UU

Element matrices:K - structural stiffnessK - dielectric permittivity C - structural dampingC - dielectric dissipationM - mass


Folded Dielectric Elastomer Actuator

• Linear Actuator– Applied voltage between

the compliant electrodes– Compresses based on

electrostatic pressure

Dielectric Elastomer

Compliant Electrode

Folded Sheet of Dielectric Elastomer

+Vin

Image courtesy of Federico Carpi, University of Pisa


Folded Dielectric Elastomer Actuator• Finite Element Model

Mechanically Fixed Base

Electrodes -8500 Volts

Actuator is 80 mm with 100 0.8 mm polymer layers.

80 m

m



• Results

Deformation Electric Potential

Deflection = 7.15 mm

Electric Potential = 8500 Volts



• Comparison with Test Data

Excellent correlation to the test data to about 7% strain. Model is stiffer than actual device between 7% and 10% strain.


Structural-Thermal-Electric Coupling• A structural-thermal-electric analysis including

thermal expansion, piezocaloric, Joule heating, Seebeck/Peltier and piezoresistive effects:

Structural Thermal

Electrical


Finite Element Formulation• Matrix/Load Vector Coupling

• U and T are strongly coupled by thermoelastic coupling matrices KUTand CTU

• T and V are strongly coupled by Seebeck coefficient matrix KVT and weakly coupled by Peltier and Joule load vectors QJ QP

• U and V are weakly coupled by the piezoresistive coefficients KVV(U)

• Unsymmetric Matrix

UU UT UU UUT J P

TT 0 UT TT

VT VV VV

K K 0 U C 0 0 U M 0 0 U F 0 K 0 T -T K C 0 T 0 0 0 T Q+Q +Q0 K K (U) V 0 0 C V 0 0 0 V I

⎧ ⎫ ⎧ ⎫⎧ ⎫ ⎧ ⎫⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎩ ⎭ ⎩ ⎭⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎩ ⎭ ⎩ ⎭

& &&

& &&

& &&

UU

TT

UT

VV

VT

UU

TT

Element matrices:K - structural stiffnessK - thermal conductivityK - thermoelastic couplingK - electric conductivity K - Seebeck coupling C - structural dampingC - specific heat dampingCVV

UU

0

- dielectric dampingM - mass T - absolute reference temperature


Thermal-Electric Actuator Example

• Thermal-Electric Actuator

SEM image courtesy of Victor Bright, University of Colorado at Boulder

Current Density

15 Volt

0 Volt

Temperature

Deflection

Deflection is induced by the differential thermal expansion between the thin and wide arms.


Conclusions

• Direct Coupled Field Elements– Advantages

• Ease of use– One element type– One finite element model– One set of results

• Robust for nonlinear coupled-field solutions• Nonlinear geometric effects

– Disadvantages• Large model sizes• Can create unsymmetric matrices

ThermalFluidsEmag

Structural

DOF: UX, UY, UZ, TEMP, VOLT, (AZ)

Documents

Multiphysics Simulation using Direct Coupled-Field … · Multiphysics Simulation using Direct Coupled-Field Element Technology Stephen Scampoli ANSYS, Inc. NAFEMS 2020 ... • A